**5. Graphical and Numerical Analysis**

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In this segment, graphical and numerical analysis is made on the solutions of resulting nonlinear ordinary differential equations mentioned in Equations (28)–(36). The differential transformation scheme is applied to present the solutions of the foregoing equations. Our principal focus is to inspect the physical characteristics of numerous physical parameters in the momentum equation, induced MHD equations, temperature distribution, motile microorganism density function, and mass transfer equation. For instance, the influence of squeezing and Rotational Reynolds number *SQ*, *R*Ω, Reiner-Rivlin fluid parameter *K*, Brownian motion *Tb*, magnetic Reynolds number Re*M*, Prandtl number *Pt*, thermophoresis parameter *Tt*, Schmidt number *SM*, Bioconvection number *Bs*, and Peclet number *Pl* are examined.

Table 1 shows the numerical comparison with previous results [56] against the torque values at the upper and the lower plate by taking *K* = 0, *Rd* = 0, *σ* = 0 in the present results. It is found that the results obtained in the present study are not only correct but also converge rapidly. Furthermore, we can also say that the proposed methodology, i.e., DTM-Padé shows promising results against the coupled nonlinear different equations.

Tables 2–4 shows the different physical parameters developed against Sherwood number, Nusselt number, and motile density function [*φ*(0), *<sup>θ</sup>*(0), *<sup>χ</sup>*(0)]. Moreover, the torque values at the lower plate *dg*(0)/*dλ*, and upper plate *dg*(1)/*dλ* are also calculated numerically in Tables 5 and 6.

**Table 1.** Comparison of the torque values at the lower and upper plate with previous results [56] when the fluid behaves as a Newtonian model (*K* = 0) and the remaining values are *R*Ω = 0.3, *FT* = 0.5, *Bt* = 0.6, *K* = 0, *Rd* = 0, *σ* = 0 for various values of *SQ* and *R*Ω.


**Table 2.** Analysis of Nusselt number *<sup>θ</sup>*(0), for multiple values *Tt*, *Tb*, *Pt*, *SQ* by DTM-Padé [5 × 5].


**Table 3.** Analysis of Sherwood number *φ*(0) for various values *Tt*, *Tb*, *SQ*, *SM*, *E*, *σ* by DTM-Padé [5 × 5].


 *K* **= 0** *K* **= 0.1** *Tt Tb SQ SM E σ* **DTM-Padé DTM-Padé** 3 −1.829718893888726 −1.8397573168178392 4 −1.821048746320453 −1.8311125640363681 1 2 −2.01093920918129 −2.020462840246665 4 −2.19121593549506 −2.2002581288517207 6 −2.359245916149752 −2.367866612938209

**Table3.***Cont.*

**Table 4.** Analysis of *χ*(0) for various values of *SQ*, *Bs*, *Pl* by DTM-Padé [5 × 5].


**Table 5.** Numerical computations of Torque at a fix circular and upper circular plates by DTM-Padé [5 × 5] for various values of Squeezing Reynolds Number *SQ*.


**Table 6.** Numerical computations of Torque at a fix circular and upper circular plates by DTM-Padé [5 × 5] for multiple values of Rotational Reynolds Number *R*Ω.


Figure 2 illustrates the influence of the velocity profile in the axial direction *f* because of the squeezed Reynolds number *SQ*, rotational Reynolds number *R*Ω, and the material parameter of Reiner-Rivlin fluid *K*. From Figure 2 one can perceive that increasing the squeezed Reynolds number *SQ* axial velocity decreases, but increasing rotational Reynolds number *R*Ω, the axial velocity profile increases. The physical reason behind this is that when we increase the value of Squeezing Reynolds number *SQ*, the distance between the plates increases, the fluid velocity decreases, and the fluid accelerates by rotation of the plate when we increase the values of rotational Reynolds number *R*Ω. Figure 3 depicts that increasing the values of the material parameter of the Reiner-Rivlin fluid increases the velocity distribution against axial direction *f* .

**Figure 2.** Implications of *SQ* and *R*Ω on velocity distribution (axial) *f* (*λ*).

**Figure 3.** Implications of *K* on velocity distribution (axial) *f* (*λ*).

Figure 4 depicts the influence of squeezing Reynolds number *SQ* and Rotational Reynolds Number *R*Ω against tangential velocity distribution *g*. From Figure 4, it can be ascertained that by enhancing the values of the squeezed Reynolds number *SQ*, the tangential velocity distribution decreases. Similar phenomena are observed in Figure 5, i.e., by increasing the values of the rotational Reynolds number, the tangential velocity profile declines.

**Figure 4.** Implications of *SQ* on velocity distribution (tangential) *<sup>g</sup>*(*λ*).

**Figure 5.** Implications of *R*Ω on velocity distribution (tangential) *<sup>g</sup>*(*λ*).

From Figure 6, it can be seen that by increasing the values of magnetic Reynolds number Re*M*, the tangential and axial magnetic field decreases, as the magnetic Reynolds number is the ratio of fluid flux to the mass diffusivity. So, by increasing the magnetic Reynolds number, a decrease in mass diffusivity and increase in fluid flux is seen. This decline in mass diffusivity disrupts the diffusion of the magnetic field and resulting, a decline in axial and tangential induced magnetic fields is observed.

**Figure 6.** Implications of Re*M* on *<sup>m</sup>*(*λ*), *n*(*λ*) in axial and tangential direction.

Figure 7 elucidates the consequences of the Brownian motion parameter and thermophoresis parameter *Tb*, *Tt* on the temperature field *θ* . The graph shows that intensifying the values of thermophoresis, Brownian motion parameter *Tt*, *Tb* increases the temperature profile. The physical reason is that the fluid temperature increases due to strengthening the kinetic energy of nanoparticles. The effects of squeezing Reynolds number *SQ* and Prandtl number *Pt* on temperature profile *θ* is displayed in Figure 8. One can notice that by enhancing the Prandtl number *Pt* and the squeezing Reynolds number *SQ*, the temperature profile *θ* diminishes. When the thermal conductivity reduces by intensifying the values of the Prandtl number *Pt* then the temperature profile *θ* declines. The effects of radiation parameter *Rd* on temperature profile *θ* are shown in Figure 9. It is observed that by enhancing the radiation parameter *Rd* the temperature profile *θ* increases. The physical reason behind this is that an increase in radiation releases the heat energy from flow; hence there is an increase in temperature.

**Figure 7.** Implications of *Tt* and *Tb* on temperature function *θ* (*λ*).

**Figure 8.** Implications of *SQ* and *Pt* on temperature function *θ* (*λ*).

**Figure 9.** Implications of *Rd*, on temperature function *θ* (*λ*).

Figure 10 shows the consequences of thermophoresis parameter *Tt* and Brownian motion *Tb* on nanoparticle concentration *φ*. It is perceived that nanoparticle concentration declines by increasing the values of Brownian motion *Tb*, and concentration of nanoparticle intensifies by increasing values of thermophoresis parameter *Tt*. In fact, gradual growth in *Tb* increases the random motion and collision among nanoparticles of the fluid, which produces more heat and eventually it results in a decrease in the concentration field. Due to increasing values of *Tt*, more nanoparticles are pulled towards the cold surface from the hot one, which ultimately results in increasing the concentration distributions. Figure 11 shows the consequences of Schmidt number *SM* and squeezed Reynolds number *SQ* on nanoparticle concentration. By enlarging the values of squeezed Reynolds number *SQ*, nanoparticle concentration *φ* increases, on the other hand, converse phenomena are noticed by enhancing the values of Schmidt number *SM*. Figure 12 deliberates the influence of reaction rate *σ* and activation energy *E* on the nanoparticle concentration *φ*. It may be observed that nanoparticle concentration displays a substantial rise by increasing values of *E*. Since high energy activation and low temperatures impart to a constant reaction rate, the

resulting chemical reaction is therefore slowed down. Consequently, the concentration of the solute rises. On the other side, by increasing values of *σ*, the nanoparticle concentration decreases.

**Figure 10.** Implications of *Tt* and *Tb* on concentration function *φ*(*λ*).

**Figure 11.** Implications of *SQ*, *SM* on concentration function *φ*(*λ*).

**Figure 12.** Implications of *SQ*, *SM* on concentration function *φ*(*λ*).

Figure 13 portrays the consequences of Peclet number *Pl* and squeezed Reynolds number *SQ* on motile microorganism density function *χ*. One can experience that enhancing values of squeezed Reynolds number *SQ* tends to boost the microorganism density function, while increasing the values of Peclet number *Pl*, the motile microorganism density function diminishes. The reason behind this is that the diffusivity of the microorganism reduces, then the speed of the microorganism also decreases. This is the physical fact and resulting in the microorganism density function decreasing while increasing the value of Peclet number *Pl*. Figure 14 is plotted to see the physical performance of the Bioconvection Schmidt number *Bs*. It is apparent that by enhancing values of bioconvection Schmidt number *Bs* the motile microorganism density function rises, but the consequences are negligible.

**Figure 13.** Implications of *SQ*, *Pl* on motile microorganism density function *<sup>χ</sup>*(*λ*).

**Figure 14.** Implications of *Bs* on motile microorganism density function *<sup>χ</sup>*(*λ*).
